Coincidences of composites of u.s.c. maps on h-spaces and applications

Abstract

Applications of the classical Knaster - Kuratowski - Mazurkiewicz (si- mply, KKM) theorem and the fixed point theory of multifunctions defined on convex subsets of topological vector spaces have been greatly improved by adopting the concept of convex spaces due to Lassonde [LIJ. In this direction, the first author [P5J found that certain coinci­ dence theorems on a large class of composites of upper semicontinuous multifunctions imply many fundamental results in the KKM theory. On the other hand, the concept of convex spaces was extended by Horvath [HI-5J to spaces having certain families of contractible subsets or H -spaces. In this direction, a number of authors extended important results on convex spaces to those on H -spaces. See Bardaro and Cep­ pitelli [BCI-3J, Ding et al. [Di, DKTI-2, DTrJ, Park [PI-3]' Tarafdar [T], and H. Kim [KJ. In the present paper, we extend the main coincidence theorem in [P5J to H-spaces and apply it to obtain a far-reaching generalization of the KKM theorem, fixed point or coincidence theorems for H -spaces, and other results. We also obtain open-valued versions of a KKM theorem and a coincidence theorem for H-spaces. Many of the main results in the above-mentioned papers are extended and unified. Especially, the main theorems of [DiJ are improved and corrected.